11.12.2009 - BIRS: Noise, Time Delay and Balance Control

Tamas Insperger

• System under consideration
  • \dot{x} = Ax + Bu
  • y = Cx
    • y is only known by y(t-\tau)
  • u = f(r-y)
    • u = Dy(t-\tau)
  • \ddot{x}(t) +a_1\dot{x}(t)+a_0x(t)=u(t)
    • linear approximation u(t)=px(t-\tau)+p\tau\dot{x}(t-\tau)
    • secant approximation u(t)=2px(t-\tau)+p\tau x(t-2\tau)
    • integral approximation u(t)=\int_0^{t-\tau} w_0 x(s) + w_1\dot{x}(s) ds
• Smith predictor
  • x = \frac{1}{s-a}u
  • s-a+b e^{\tau s} or \dot{x}(t) + a x(t) - b x(t-\tau)
  • becomes \ddot{x}(t) +(b-2a)\dot{x}(t) + a(a-b)x(t) if estimated delay and state are same as the real delay and state
  • Smith predictor is sensitive to parameters and not necessarily good to stabilize unstable systems
• Predictive control
  • Use the available delayed state to predict the current state
• Delayed feedback
  • Use the delayed output directly
• Classification of linear systems
  • Time invariant ODE
    • n dimensions = n eigen-values
  • Time invariant DDE
    • n dimensions = infinite number of eigen-values
  • Time periodic ODE
    • On one period (need to know the map) then the there are n eigen-values for n-dimensions of the period
    • \dot{x}(t)=A(t)x(t)
    • A(t+T) = A(t)
    • x(T) = \Phi x(0)
  • Time periodic DDE
    • Infinite number of eigen-values
• Brockett problem
  • \dot{x}(t) = A x(t) + BG(t)Cx(t-\tau)
• Act and wait
  • G(t) = 0 during waiting period, which is longer than delay
  • G(t) = gg(t) during acting period, which is shorter than delay
  • Step-by-step solution
    • \dot{x}(t)=Ax(t)
    • x(t)=\Phi^1(t)x(0)
    • \Phi^1(t)=e^{At}
    • \Phi^2(t)=e^{At}+\int_{tw}^t e^{A(t-s)} B \Gamma(s) C e^{A(s-/tau)} ds
    • \Phi^3(t)=\Phi^2(t)+stuff
    • x(T)=\Phi^3(T)x(0)
      • This ends up with in n-poles.

Francisco Valero-Cuevas

• Internal models
• Predictive strategies
• State estimators
• Development in childhood
• Biological computation

James Finley and Eric Perreault

• Feedforward vs feedback control during balance
• Heightened co-contraction (Hogan 1984; Milner 2002)
• Larger gain on stretch reflexes during a "compliant" environment
• Co-contraction strategy
  • see co-contraction increase in "unstable" condition
  • co-contraction appears to inhibit stretch reflex
  • Ankle stiffness increases as stability of joint decreases
  • What about signs on net stiffness? -Kank-Kenv+mgl
• There appears to be a bias towards feedforward over feedback control

Tim Kiemel and John Jeka

• Linearized "unstable" pendulum model with delayed PD control
  • Plant is the mapping from EMG to body segment angles
  • Feedback is the mapping between changes in body angles into EMG
  • Intrinsic stiffness vs ankle stiffness
    • stability achieved with a combination of hip and ankle stiffness
  • Feedback in the nervous system is probably not PD, most likely something better

Meeting thoughts

• Intermittent vs Continuous and Linear vs Non-linear
  • How do these fundamental questions help/hurt modeling of posture?
• Kleinman DL, "Optimal control with time delay and observation noise." IEEE Trans Automatic Control (15)524-527, 1969
• Palmor ZJ (1996) Time delay compensation smith predictor and its modifications. Levine "The control handbook", CRC Press

Gabor Stepan

• Chaos is amusing. This means that there needs to be large non-linearities
• Digital control systems introduce "spatial" and "temporal" delays.
• There is a point where there is "micro-chaos" due digitization where the system is caught in an oscillation before it gets to the regulated point.
• Are the delays in the human system a continuous delay or a discrete delay?
• Delayed feedback on jerk!

Questions for future math problems

• Instead of linearizing equations that have noise in them transfer the system into probability space where the equations are linear in probability
  • Delayed Langevin equation
• Try and determine if you can tell if a system is linear or non-linear from time-series data